Re: P_{n} set of all polynomials of degree n

In the induction step, you need to show that the Cartesian product of two countable sets (the set of the new leading coefficients, which is , and the set of polynomials of the previous degree, which is countable by the induction hypothesis) is countable. See this thread. It contains the exact formula for the bijection, but you may not need it.

Assume P(k) is true for a fixed but arbitrary

What is n here?

Originally Posted by dwsmith

Assume P(k) is true for a fixed but arbitrary , where P(k) is defined as

First, this is not a good definition because have not been defined before. Is P(k) a fixed polynomial or a set of polynomials? In either of those cases, P(k) cannot be true or false.